The hyperbolic lines of finite symplectic spaces
type: Item
instance of: scholarly article
title: The hyperbolic lines of finite symplectic spaces
zbMATH Open document ID: 0643.51012
DOI: 10.1016/0097-3165(88)90022-2
published in: Journal of Combinatorial Theory. Series A
publication date: 1988-01-01T00:00:00Z
review text: Let (\({\mathcal P},{\mathcal L})\) be a finite, connected partial linear space in which each pair of intersecting lines lies in a subspace which is a projective plane minus a point and all lines through it. Assume that (\({\mathcal P},{\mathcal L})\) contains at least two such planes. The author proves that either (1) (\({\mathcal P},{\mathcal L})\) is a symplectic space, i.e. for some prime power q and some integer d at least 3, (\({\mathcal P},{\mathcal L})\) is isomorphic to the partial linear space of hyperbolic lines of a symplectic geometry of the Desarguesian projective space PG(d,q); or (2) all lines of \({\mathcal L}\) contain exactly three points and so (\({\mathcal P},{\mathcal L})\) is one of the spaces discussed in a previous article by the author. The spaces in (2) are not necessarily symplectic, but all can be embedded in symplectic spaces. The proof involves the consideration of a collineation group of (\({\mathcal P},{\mathcal L})\) generated by elations.
Mathematics Subject Classification ID: 51E30
zbMATH DE Number: 4048519
zbMATH Keywords: partial linear space
zbMATH Keywords: symplectic spaces
author: J. I. Hall
MaRDI profile type: Publication
full work available at URL: 0097-3165(88)90022-2
OpenAlex ID: W2086939732
cites work: Q3708174
cites work: Q5817043
cites work: Some groups generated by transvection subgroups
cites work: Q5545191
cites work: Dual affine geometries and alternative bilinear forms
cites work: Q4107146
cites work: CLASSIFYING COPOLAR SPACES AND GRAPHS
cites work: Graphs, Geometry, 3-Transpositions, and Symplectic F 2 -Transvection Groups
cites work: Some 3-Transposition Groups with Normal 2-Subgroups
cites work: A characterization of finite symplectic groups over fields with more than two elements
cites work: Geometries with dual affine planes and symplectic quadrics
cites work: Q4154874
description: scientific article; zbMATH DE number 4048519